📐🔗 TM 1 – Set Theory + Binary Relations

📅 Exam Details

📚 Coverage

The first theoretical minimum covers fundamental concepts from the initial modules of the course:

📐 Set Theory (Weeks 1–2, 6)

• Set operations and laws
• Power sets
• Cartesian products
• Russell’s paradox
• Axiomatic set theory (ZFC)
• Cardinality
• Countable vs uncountable sets
• Cantor’s diagonalization
• Schroeder-Bernstein theorem

🔗 Binary Relations (Weeks 3–7)

• Relation properties (reflexive, symmetric, transitive, antisymmetric)
• Equivalence relations and partitions
• Order relations and Hasse diagrams
• Functions (injection, surjection, bijection)
• Composition and inverses
• Lattices

📝 Sample Questions

Definitions

• Define equivalence relation (with example)
• What is a well-ordering?
• Define power set and its cardinality
• Define bijection
• What is a partition?

Theorems

• State equivalence-partition correspondence theorem
• State Cantor’s theorem
• State Schroeder-Bernstein theorem
• Composition associativity

Proofs

• Prove real numbers are uncountable
• Show composition of bijections is bijective
• Prove every finite poset has maximal element

Conceptual

• Why does Russell’s paradox matter?
• Difference between maximal and greatest element?
• Example of injective but not surjective function
• Why are rationals countable but reals uncountable?

✅ What You Must Know

• Set operations (union, intersection, difference, symmetric difference) and laws
• Relation properties (reflexive, symmetric, transitive, antisymmetric)
• Equivalence relations and partitions
• Partial orders, total orders, well-orders
• Injective, surjective, bijective functions
• Composition of relations and functions
• Cardinality (finite, countable, uncountable)
• Key theorems: Cantor’s theorem, Schroeder-Bernstein, equivalence-partition correspondence
• Proof techniques: element method, double inclusion, diagonalization

📖 Preparation Checklist

✅ Week 6 (2 weeks before)

• Review all lecture notes from Weeks 1–7
• Create flashcards for definitions
• List all theorem statements
• Identify 5 hardest concepts

✅ Week 7 (1 week before)

• Practice 10+ proofs without looking
• Join study group, quiz each other
• Can recite all definitions precisely?
• Attend review session

✅ Day Before

• Light review of materials
• Get 8 hours sleep!
• Prepare mentally, stay calm

💡 Pro Tips

🎯 Success Strategy

• Definitions first: If you can’t define it, you can’t use it
• Proof structure matters: Introduction → Body → Conclusion
• Use examples: Verify your reasoning with concrete cases
• Time management: Don’t spend 60 min on one proof
• Partial credit: Write what you know, even if incomplete

Common Pitfalls to Avoid

• ❌ Confusing “injective” and “surjective”
• ❌ Writing “obvious” instead of proving rigorously
• ❌ Circular reasoning in proofs
• ❌ Mixing up “element of” and “subset of”
• ❌ Forgetting to check all relation properties

What Graders Look For

• ✓ Precise definitions (exact wording)
• ✓ Complete proofs (no gaps in logic)
• ✓ Clear explanations (not just symbols)
• ✓ Correct examples and counterexamples
• ✓ Proper mathematical notation